Baily's paradox

1

Richard P. Brief NEW YORK UNIVERSITY
BAILY'S PARADOX
The following question appeared in Francis Baily's celebrated 1808 text on interest and annuities: "If a penny had been put out at 5 percent compound interest at the birth of Christ; to what sum would it amount at the end of the year 1810?" . Baily gave this solution :
By the first theorem ... it will be seen that its amount in that time would be .004166666 x (1.05)1810 = 938469 000000 000000 000000 000000 000000 pounds. Now the diameter of the earth is about 8000 miles; consequently its solid contents* will be 68 188963 498145 531559 936000 cubic inches: and if it were made of standard gold, each cubic inch being worth 38 /. 10 s.**, the total value of such a globe would be 2625 275094 678602 965057 536000 pounds. But the amount of a penny in 1810 years as above stated, is more than 357 474 600 times the value of such a globe: consequently if one penny had been put out at 5 percent compound interest at the time above men-tioned, it would, at this period, have amounted to more money than could be expressed by THREE HUNDRED AND FIFTY-SEVEN MILLIONS of globes, each equal to the Earth in magnitude, and all solid gold!!! Whereas if it had been put out at the same rate of simple interest, the amount in the same time would have been only seven shillings and seven-pence half-penny.
Baily's calculation is "slightly" different than the 357,426,300 globes determined by my own calculation. Even allowing for a pos-sible error, perhaps someone might want to consider planning for
*To find the solid contents of a sphere, multiply the decimal .785398 163397 448309 615&C, by the cube of the diameter, and take two-thirds of the product.
**Since a cubic inch of distilled water weighs about 254 grains, and the specific gravities of standard gold and water are to each other as 18.888 to one, it follows that a cubic inch of gold, at the mint price of 3 l. 17 s. 10½ d. per ounce, will amount to 38 l. 10 s. d.